Quote:
Originally Posted by GeeTwo
What is the average length of all the line segments which can be drawn within the unit circle (1 unit in radius, 2 units in diameter)?
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This can actually be solved in closed form with much less intricate calculus than the case of the square, with proper selection of coordinate system.
Some warmup questions to this if you can't figure out where to start:
- What is the average of the squares of the lengths of all the line segments which can be drawn within the unit circle? [taking out the square root simplifies the problem greatly, and helps provide an upper bound to the original question]
- What is the average of the lengths of all the chords of the unit circle? (That is, line segments with both endpoints on the border of the circle)
- What is the average of the lengths of all the line segments between a point on the [border of] a circle of radius R, and a point within the same circle?
OBTW, with proper scaling, Greg's answer for the 100000-gon of area 1 provides an answer to the original problem good to within 1 part per 10,000:
Quote:
Originally Posted by Greg Woelki
Here is a generalized Monte Carlo simulation ....
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And, OBTW2: if you can find the closed form, each of the four questions can be calculated in fewer than 10 keystrokes on a calculator with the following buttons:
Quote:
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0 1 2 3 4 5 6 7 8 9 + - / * π √ =
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